American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy
  • EECT Home
  • This Issue
  • Next Article
    Periodic solutions and multiharmonic expansions for the Westervelt equation
June  2021, 10(2): 249-258. doi: 10.3934/eect.2020064

Decay rate of global solutions to three dimensional generalized MHD system

Yanxia Niu 1, , Yinxia Wang 2,, and  Qingnian Zhang 2,

1. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
2. 

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

* Corresponding author: Yinxia Wang

Received  September 2019 Published  June 2020

Fund Project: The second author is supported in part by the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No. 20ZX002)

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space $ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $ follows by constructing time weighted energy inequality.

Keywords: generalized MHD system, decay rate of global solutions, analyticity.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35B40.
Citation: Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1. Google Scholar

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.  Google Scholar

[12]

Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

W. Wang, T. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.  Google Scholar

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

show all references

References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1. Google Scholar

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.  Google Scholar

[12]

Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

W. Wang, T. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.  Google Scholar

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

[1]

Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036
[2]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[3]

Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
[4]

Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240
[5]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[6]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[7]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[8]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337
[9]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008
[10]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[11]

Shikuan Mao, Yongqin Liu. Decay of solutions to generalized plate type equations with memory. Kinetic & Related Models, 2014, 7 (1) : 121-131. doi: 10.3934/krm.2014.7.121
[12]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433
[13]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713
[14]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071
[15]

Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic & Related Models, 2014, 7 (1) : 195-203. doi: 10.3934/krm.2014.7.195
[16]

Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307
[17]

Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007
[18]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
[19]

Kazuo Yamazaki. $(N-1)$ velocity components condition for the generalized MHD system in $N-$dimension. Kinetic & Related Models, 2014, 7 (4) : 779-792. doi: 10.3934/krm.2014.7.779
[20]

Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 513-532. doi: 10.3934/cpaa.2008.7.513

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (110)
  • HTML views (358)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences